Schedule

UNM Physics and Astronomy, Summer 2024 course

Introduction:

1. Introduction [Lu and John] [May 29]

Optimization:

1. Mathematical preliminaries and optimality conditions for unconstrained optimization [Lu, June 5th] [Lec1]
2. Gradient descent and Newton's method [Srinidhi Pawar and Samuel Goodwin, June 12th] [Lec2] [Lec2-extra] [Codes]
3. Convexity and convex optimization [Cancelled, June 18th (Tuesday)]
4. Linear programming and semidefinite programming (SDP) [Alex Fischer and Morteza Darvishi, July 3rd]

Quantum Foundations:

1. The death of local realism. And yet, the evidence for epistemics [Basie Seibert and Leroy Fagan] [July 10]
2. Foils and reconstructions [Robby Kramer and Cole Kelson-Packer] [July 17]
3. Wigner and his friends [Mohsin Raza and Andrew Forbes] [July 24]
4. Quantum Causality and Quantum Reference Frames [Ivy Gunther and Jalan Ziyad] [July 31]
5. De Finetti and the unknown [Shravan Shravan and Chaithanya Rayudu] [August 7]
6. Interpretations and the road ahead [John DeBrota] [August 14]

Detailed Schedule

Introduction:

1. Introduction [Lu and John] [May 29]

Overview of the course: course plans, logistics, and motivations.

Optimization:

• Mathematical preliminaries & optimality conditions for unconstrained optimization [Lu, June 5th]

Brief review of important notions and results from calculus, linear algebra, and topology that will be frequently used through this course. Discussion of first and second-order optimality conditions for unconstrained optimization. [Chapter 1 & 2, Beck].

• Gradient descent & Newton's method [Srinidhi Pawar and Samuel Goodwin, June 12th]

Detailed study of the two basic optimization methods: gradient descent and Newton's method. [Chapter 4 & 5, Beck].

• Convexity & convex optimization [Cancelled, June 18th]

Introduction to the most important optimization class: convex optimization.

• Linear programming & semidefinite programming (SDP) [Alex Fischer and Morteza Darvishi, July 3rd]

Introduction to the two most widely used optimization paradigms: Linear programming and semidefinite programming.

Quantum Foundations:

1. The death of local realism. And yet, the evidence for epistemics [Basie Seibert and Leroy Fagan] [July 10]



Is quantum mechanics incomplete? Einstein, Podolsky, and Rosen famously thought so; Bell famously showed they were wrong. Since then, the prospect for quantum theory to fit into the familiar physical categories of local and "realistic" has been attacked from many angles. These are exemplified by the Bell/CHSH inequalities, the Kochen-Spekker theorem, and the Pusey-Barrett-Rudolph (PBR) theorem. Evidently there are no non-contextual local hidden variables underlying quantum theory. Does this mean the quantum state real? It turns out it's not so simple. Many have persuasively argued that we should focus on epistemics (concerning information, knowledge, or beliefs) to get to the bottom of what quantum theory can tell us about reality.

2. Foils and reconstructions [Robby Kramer and Cole Kelson-Packer] [July 17]



Much of the work done in quantum foundations engages with one or both of these interrelated topics. A so-called foil theory attempts to study quantum theory by studying what it is not. Seeing what can and cannot be done in a foil may allow us to identify what is unique about quantum theory and hence our world. Meanwhile, a reconstruction of quantum theory is a set of axioms which together imply a structure isomorphic to quantum theory. The hope here is that a particularly illuminating set of axioms will cut quantum theory at instructive seams, again allowing us to highlight what is special and what is not.

3. Wigner and his friends [Mohsin Raza and Andrew Forbes] [July 24]



Wigner's famous thought experiment is a fascinating and instructive paradox. Really it is one of the key litmus tests: every interpretation of quantum theory must address it in one way or another. Briefly, Wigner stands outside of a lab which holds his friend and a spin-1/2 particle. The friend will measure the spin, obtaining an outcome. Wigner, meanwhile, considers the joint system consisting of his friend and the spin as an astronomically complex closed system and thus considers it to be evolving unitarily. For Wigner, this evolution produces an entangled pure state, but the friend obtains a spin projection. For Wigner there is no measurement outcome. Who is right? Recently there have been elaborations of Wigner's original argument, potentially allowing us to put a finer point on what these thought experiments imply about nature.

4. Quantum Causality and Quantum Reference Frames [Ivy Gunther and Jalan Ziyad] [July 31]



This lecture will be a double feature introducing two areas of exploration in foundations. In the first half, we will discuss causality. What does it mean to talk about causality in relation to quantum theory? To what extent are we forced to revise our traditional notions? What can exploring this subject teach us about the nature of reality? In the second half, we will meet the notion of a quantum reference frame. Again we ask: What does it mean? What are the main questions? And how does it deepen our understanding, both applied and foundational?

5. De Finetti and the unknown [Shravan Shravan and Chaithanya Rayudu] [August 7]



By now there are multiple quantum de Finetti theorems. What even is a de Finetti theorem? Ater motivating the idea, we will learn about de Finetti's original work and its first quantum extension. How, if at all, does quantum theory change the story? We will then explore some newer variations and discuss their foundational and applied significance.

6. Interpretations and the road ahead [John DeBrota] [August 14]



Let's take stock. We've toured some of the main attractions in quantum foundations, but conspicuously avoided the one that really gets people going: interpretations. This topic was saved for last so that we could approach it responsibly. With the previous lectures under our belts, we are hopefully now more equipped to assess some of the proposed ways to understand what quantum theory is. What does it mean and what does it not mean for the nature of reality? Some interpretations come packaged up with an ontology while others try to identify in which corners we might find one. Still other approaches are not interpretations at all, but rather proposed modifications to the formalism. I must disclose I am far from unbiased on this subject, but I will try to be fair; I don't want to indoctrinate, I want to understand. We're coming up on a century of work to demystify the quantum and it'll be a glorious day when the work is finally done.

Bibliography

Optimization Resources:

1. Amir Beck, Introduction to Nonlinear Optimization: Theory, Algorithms, and Applications with Python and MATLAB, Second Edition, SIAM, 2023. [Online book].
2. Sébastien Bubeck, Convex Optimization: Algorithms and Complexity, 2015.
3. László Lovász, Semidefinite programs and combinatorial optimization.
4. Bernd Gärtner and Jiri Matousek, Approximation Algorithms and Semidefinite Programming, Springer, 2010.
5. Carl W. Lee, Linear Programming Notes.

Foundations Resources:

• Death of realism
1. J. S. Bell, “On the Einstein Podolsky Rosen paradox”
2. J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, “Proposed Experiment to Test Local Hidden-Variable Theories”
3. S. Kochen and E. P. Specker, “The Problem of Hidden Variables in Quantum Mechanics”
4. M. F. Pusey, J. Barrett, and T. Rudolph, “On the reality of the quantum state”
5. N. D. Mermin, “Quantum Mysteries for Anyone”
6. M. S. Leifer, “Is the quantum state real? An extended review of $\psi$-ontology theorems”
• The evidence for epistemics
1. R. W. Spekkens, “Evidence for the epistemic view of quantum states: A toy theory”
2. C. A. Fuchs, “Quantum Mechanics as Quantum Information (and only a little more)”
3. M. S. Leifer, “Is the quantum state real? An extended review of \psi-ontology theorems”
4. L. Catani, M. Leifer, D. Schmid, and R. W. Spekkens, “Why interference phenomena do not capture the essence of quantum theory”
• Foils and Reconstructions
1. G. Chiribella and R. W. Spekkens, Eds., Quantum Theory: Informational Foundations and Foils, vol. 181. in Fundamental Theories of Physics, vol. 181.
2. N. Harrigan and R. W. Spekkens, “Einstein, Incompleteness, and the Epistemic View of Quantum States”
3. S. Popescu and D. Rohrlich, “Quantum nonlocality as an axiom”
4. L. Hardy, “Quantum Theory From Five Reasonable Axioms”
• Wigner and his friends
1. E. P. Wigner, “Remarks on the Mind-Body Question”
2. D. Frauchiger and R. Renner, “Quantum theory cannot consistently describe the use of itself”
3. K.-W. Bong et al., “A strong no-go theorem on the Wigner’s friend paradox”
4. D. Schmid, Y. Ying, M. S. Leifer, "A review and analysis of six extended Wigner’s friend arguments"
• Quantum Causality
1. Quanta Magazine: Quantum Mischief Rewrites the Laws of Cause and Effect
2. L. Hardy, "Probability Theories with Dynamic Causal Structure: A New Framework for Quantum Gravity"
3. Č. Brukner, "Quantum Causality"
4. O. Oreshkov, F. Costa, and Č. Brukner, "Quantum correlations with no causal order"
5. M. Araújo, C. Branciard, F. Costa, A. Feix, C. Giarmatzi, and Č. Brukner, "Witnessing causal nonseparability"
6. R. Chaves et al., "Causal Networks and Freedom of Choice in Bell’s Theorem"
7. C. J. Wood and R. W. Spekkens, “The lesson of causal discovery algorithms for quantum correlations: causal explanations of Bell-inequality violations require fine-tuning”
8. D. Schmid, J. H. Selby, R. W. Spekkens, "Unscrambling the omelette of causation and inference: The framework of causal-inferential theories"
• Quantum Reference Frames
1. S. Bartlett, T. Rudolph, and R. W. Spekkens, "Reference frames, superselection rules, and quantum information"
2. S. Bartlett, T. Rudolph, and R. W. Spekkens, "Dialogue Concerning Two Views on Quantum Coherence: Factist and Fictionist"
3. B. Schumacher and M. Westmoreland, "Interpretation of quantum theory: the quantum "grue-bleen" problem"
• De Finetti and the unknown
1. C. M. Caves, C. A. Fuchs, and R. Schack, “Unknown Quantum States: The Quantum de Finetti Representation”
2. C. A. Fuchs, R. Schack, and P. F. Scudo, “De Finetti representation theorem for quantum-process tomography”
3. F. G. S. L. Brandao and A. W. Harrow, “Quantum de Finetti Theorems under Local Measurements with Applications”
4. F. Costa, J. Barrett, and S. Shrapnel, “A de Finetti theorem for quantum causal structures”